Skip to main content
SpringerLink
Log in

On Beta-Shifts Having Arithmetical Languages

  • Conference paper
Mathematical Foundations of Computer Science 2005 (MFCS 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3618))

Abstract

Let β be a real number with 1 < β < 2. We prove that the language of the β-shift is Δ\(^{\rm 0}_{n}\) iff β is a Δ n -real. The special case where n is 1 is the independently interesting result that the language of the β-shift is decidable iff β is a computable real. The “if” part of the proof is non-constructive; we show that for Walters’ version of the β-shift, no constructive proof exists.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barnsley, M.F.: Fractals Everywhere. Morgan Kaufmann, San Francisco (1993)

    MATH  Google Scholar 

  2. Blanchard, F.: β-Expansions and Symbolic Dynamics. Theoretical Computer Science 65, 131–141 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bridges, D.S.: Computability: A Mathematical Sketchbook. In: Graduate Texts in Mathematics, vol. 146, Springer, Heidelberg (1994)

    Google Scholar 

  4. Delvenne, J.-C., Blondel, V.D.: Quasi-Periodic Configurations and Undecidable Dynamics for Tilings, Infinite Words and Turing Machines. Theoretical Computer Science 319, 127–143 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Frougny, C., Solomyak, B.: Finite Beta-Expansions. Ergodic Theory and Dynamical Systems 12, 713–723 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hansel, G., Perrin, D., Simon, I.: Compression and Entropy. In: Finkel, A., Jantzen, M. (eds.) STACS 1992. LNCS, vol. 577, pp. 515–528. Springer, Heidelberg (1992)

    Google Scholar 

  7. Hofbauer, F.: β-Shifts have Unique Maximal Measure. Monatshefte für Mathematik 85, 189–198 (1978)

    Article  MathSciNet  Google Scholar 

  8. Hurd, L., Kari, J., Culik, K.: The Topological Entropy of Cellular Automata is Uncomputable. Ergodic Theory and Dynamical Systems 12, 255–265 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  9. Koiran, P.: The Topological Entropy of Iterated Piecewise Affine Maps is Uncomputable. Discrete Mathematics and Theoretical Computer Science 4(2), 351–356 (2001)

    MATH  Google Scholar 

  10. Lind, D.: The Entropies of Topological Markov Shifts and a Related Class of Algebraic Integers. Ergodic Theory and Dynamical Systems 4, 283–300 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  12. Morse, M.: Recurrent Geodesics on a Surface of Negative Curvature. Transactions of the American Mathematical Society 22, 84–110 (1921)

    Article  MATH  MathSciNet  Google Scholar 

  13. Myhill, J.: Criteria of Constructivity for Real Numbers. Journal of Symbolic Logic 18, 7–10 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  14. Odifreddi, P.: Classical Recursion Theory. Studies of Logic and the Foundations of Mathematics, vol. 129. North-Holland, Amsterdam (1989)

    MATH  Google Scholar 

  15. Parry, W.: On the β-Expansion of Real Numbers. In: Acta Math. Acad. Sci. Hung, pp. 401–416 (1960)

    Google Scholar 

  16. Renyi, A.: Representations for Real Numbers and their Ergodic Properties. Acta Math. Acad. Sci. Hung. 8, 477–493 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  17. Rice, H.G.: Recursive Real Numbers. Proceedings of the American Mathematical Society 5, 784–791 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  18. Rogers Jr., H.: Theory of Recursive Functions and Effective Computability. paperback edition. The MIT Press, Cambridge (1987)

    Google Scholar 

  19. Schmidt, K.: On Periodic Expansions of Pisot Numbers and Salem Numbers. Bulletin of the London Mathematical Society 12, 269–278 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  20. Sidorov, N.: Almost Every Number has a Continuum of Beta-Expansions. American Mathematic Monthly 110, 838–842 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Sidorov, N.: Arithmetic Dynamics. In: Topics in Dynamics and Ergodic Theory. London Mathematical Society Lecture Notes Series, vol. 310, pp. 145–189. London Mathematical Society (2003)

    Google Scholar 

  22. Takahashi, Y.: Shift with Free Orbit Basis and Realization of One-Dimensional Maps. Osaka Journal of Mathematics 20, 599–629 (1983)

    MATH  MathSciNet  Google Scholar 

  23. Turing, A.M.: On Computable Numbers with an Application to the, Entscheidungsproblem. Proceedings of the London Mathematical Society 42(2), 230–265 (1936)

    MATH  Google Scholar 

  24. Walters, P.: An Introduction to Ergodic Theory. In: Graduate Texts in Mathematics, vol. 79, Springer, Heidelberg (1981)

    Google Scholar 

  25. Weihrauch, K.: Computable Analysis: An Introduction. Springer, Heidelberg (1998)

    Google Scholar 

  26. Zheng, X.: Recursive Approximability of Real Numbers. Mathematical Logic Quarterly 48, 131–156 (2002)

    MATH  Google Scholar 

  27. Zheng, X., Weihrauch, K.: The Arithmetical Hierarchy of Real Numbers. Mathematical Logic Quarterly 47(1), 51–65 (2001)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Copenhagen (DIKU), Universitetsparken 1, DK-2100, Copenhagen Ø, Denmark

    Jakob Grue Simonsen

Authors
  1. Jakob Grue Simonsen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Mathematics, Gdansk University, Wita Stwosza 57, 80 952, Gdansk, Poland

    Joanna Jȩdrzejowicz  & Andrzej Szepietowski  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Simonsen, J.G. (2005). On Beta-Shifts Having Arithmetical Languages. In: Jȩdrzejowicz, J., Szepietowski, A. (eds) Mathematical Foundations of Computer Science 2005. MFCS 2005. Lecture Notes in Computer Science, vol 3618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11549345_65

Download citation

  • DOI: https://doi.org/10.1007/11549345_65

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28702-5

  • Online ISBN: 978-3-540-31867-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation